Email this article 
ON THE CLASSIFICATION OF FUNCTION GERMS OF TWO VARIABLES THAT ARE EQUIVARIANT SIMPLE WITH RESPECT TO AN ACTION OF THE CYCLIC GROUP OF ORDER THREE
- Authors: Astashov E.A.1
-
Affiliations:
- post-graduate student, Lomonosov Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia.
- Issue: Vol 22, No 3-4 (2016)
- Pages: 7-13
- Section: Articles
- URL: https://journals.ssau.ru/est/article/view/4252
- DOI: https://doi.org/10.18287/2541-7525-2016-22-3-4-7-13
- ID: 4252
Cite item
Full Text
Abstract
We consider the problem to classify function germs (C2 , 0) → (C, 0), that are equivariant simple with respect to nontrivial actions of the group Z3 on C2 and on C up to equivariant automorphism germs (C2 , 0) → (C2 , 0). The complete classification of such germs is obtained in the case of nonscalar action of Z3 on C2 that is nontrivial in both coordinates. Namely, a germ is equivariant simple with respect to such a pair of actions if and only if it is equivalent to ine of the following germs:
- (x, y) → x3k+1 + y2 , k ≥ 1;
- (x, y) → x2y + y3k−1 , k ≥ 2;
- (x, y) → x4 + xy3
- (x, y) → x4 + y5 .
About the authors
E. A. Astashov
post-graduate student, LomonosovMoscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia.
Author for correspondence.
Email: morenov.sv@ssau.ru
References
- Arnold V.I. Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek and Lagrangian singularities. Functional Anal. Appl., 1972, vol. 6, no. 4, pp. 254–272. .
- Arnold V.I. Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Russian Math. Surveys, 1979, vol. 34, no. 1, pp. 1–42. .
- Domitrz W., Manoel M., Rios P. de M. The Wigner caustic on shell and singularities of odd functions. J. of Geometry and Physics, 2013, vol. 71, pp. 58–72 .
- Astashov E.A. On the classification of singularities that are equivariant simple with respect to representations of cyclic groups. Bulletin of Udmurt University. Mathematics, Mechanics, Computer Science, 2016, vol. 26, no. 2, pp. 155-159 .
- Bruce J.W., Kirk N.P., du Plessis A.A. Complete transversals and the classification of singularities. Nonlinearity, 1997, vol. 10, pp. 253–275 .
- Arnold V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of differentiable maps, Volumes 1 and 2. Boston: Birkhauser, Monographs Math., 1985–1988, Vol. 82–83, 845 p. .
Supplementary files
